The Mystery of Tau – Miskolczi – Part Three – Kinetic Energy

In Part One we looked at the calculation of total atmospheric optical thickness.

In Part Two we looked at the claim that the surface and atmosphere exchanged exactly equal amounts of energy by radiation. A thermodynamics revolution if it is true, as the atmosphere is slightly colder than the surface. This claim is not necessary to calculate optical thickness but is a foundation for Miskolczi’s theory about why optical thickness should be constant.

Regarding the origin, EU is more closely related to the total internal kinetic energy of the atmosphere, which – according to the virial theorem – in hydrostatic equilibrium balances the total gravitational potential energy. To identify EU as the total internal kinetic energy of the atmosphere, the EU = SU / 2 equation must hold.

Many people have puzzled over the introduction of the virial theorem (note 1), which relates total kinetic energy of the atmosphere to total potential energy of the atmosphere. Generally, there is a relationship between potential energy and kinetic energy of an atmosphere so I don’t propose to question it, we will accept it as a given.

By the way, on the diagram SU = SG, i.e. SU = upwards radiation from the surface. And EU = upwards radiation from the atmosphere (cooling to space).

Kinetic Energy of a Gas

For people who don’t like seeing equations, skip to the statement in bold at the end of this section.

As an example of the form we might expect, if we had a very opaque atmosphere (in longwave), then EU = σTA4 (the Stefan-Boltzmann equation for thermal radiation). As the emissivity of the atmosphere reduces then the equation won’t stay exactly proportional to the 4th power of temperature. But it can never be linearly proportional to temperature.

A Mystery Equation

Many people have puzzled over the equations in Miskolczi’s 2007 paper.

Note that I have added a prefix to the equation numbers to identify they as Miskolczi’s. As previously commented, the P term (geothermal energy) is so small that it is not worth including. We will set it to zero and eliminate it, to make it a little easier to see the problems. Anyone wondering if this can be done – just set F’ = F0 + P0 and replace F0 with F’ in the following equations.

So:

EU = F + K (M5a)
SU − F0 = ED − EU (M6a)

Please review figure 1 for explanation of the terms.

If we accept the premise that AA = ED then these equations are correct (the premise is not correct, as shown in Part Two).

M5a is simple to see. Taking the incorrect premise that surface radiation absorbed in the atmosphere is completely re-emitted to the surface: therefore, the upward radiation from the atmosphere, EU must be supplied by the only other terms shown in the diagram – convective energy plus solar radiation absorbed by the atmosphere.

What about equation M6a? Physically, what is the downward energy emitted by the atmosphere minus the upward energy emitted by the atmosphere? What is the surface upward radiation minus the total solar radiation?

Well, doesn’t matter if we can’t figure out what these terms might mean. Instead we will just do some maths, using the fact that the surface energy must balance and the atmospheric energy must balance.

This just says that Surface radiation absorbed in the atmosphere + convection from the surface to the atmosphere + absorbed solar radiation in the atmosphere = energy radiated by the atmosphere from the top and bottom.

Given the (incorrect) premise that AA = ED, we can rewrite equation 10:

K + F = EU [10a]

We can see that this matches M5a, which is correct, as already stated.

So first, let’s write down the surface energy balance:

F0 – F + ED = SU + K [11]

This just says that Solar radiation absorbed at the surface + downward atmospheric radiation = surface upward radiation + convection from the surface to the atmosphere.

Please review Figure 1 to confirm this equation.

Now let’s rewrite equation 11:

SU – F0 = ED – F – K [11a]

and inserting eq 10a, we get:

SU – F0 = ED -EU [11b]

Which agrees with M6a.

And as an aside only for people who have spent too long staring at these equations – re-arrange the terms in 11b:

Su – Ed = F0 – Eu; The left side is surface radiation – absorbed surface radiation in the atmosphere (accepting the flawed premise) = transmitted radiation. The right side is total absorbed solar radiation – upward emitted atmospheric radiation. As solar radiation is balanced by OLR, the right side is OLR – upward emitted atmospheric radiation = transmitted radiation.

Now, let’s see the mystery step :

In Eq. (6) SU − (F0 + P0 ) and ED − EU represent two flux terms of equal magnitude, propagating into opposite directions, while using the same F0 and P0 as energy sources. The first term heats the atmosphere and the second term maintains the surface energy balance. The principle of conservation of energy dictates that:
SU − (F0) + ED − EU = F0 = OLR (M7)

This equation M7 makes no sense. Note that again I have removed the tiny P0 term.

Let’s take [11b], already demonstrated (by accepting the premise) and add (ED -EU) to both sides:

SU – F0 + (ED – EU) = ED – EU+ (ED -EU) = 2(ED -EU) [12]

So now the left side of eq 12 matches the left side of M7.

The M7 equation can only be correct if the right side of eq 12 matches the right side of M7:

2(ED -EU) = F0 [13] – to be confirmed or denied

In concept, this claim is that downward radiation from the atmosphere minus upward radiation from the atmosphere = half the total planetary absorbed solar radiation.

I can’t see where this has been demonstrated.

It is not apparent from energy balance considerations – we wrote down those two equations in [10] and [11].

We can say that energy into the climate system = energy out, therefore:

Which doesn’t move us any closer to the demonstration we are looking for.

Perhaps someone from the large fan club can prove equation 7. So many people have embraced Miskolczi’s conclusion that there must be a lot of people who understand this step.

Conclusion

I’m confused about equation 7 of Miskolczi.

Running with the odds, I expect that no one will be able to prove it and instead I will be encouraged to take it on faith. However, I’m prepared to accept that someone might be able to prove that it is true (with the caveat about accepting the premise already discussed).

The more important point is equating the kinetic energy of the atmosphere with the upward atmospheric radiation.

It’s a revolutionary claim.

But as it comes with no evidence or derivation and would overturn lots of thermodynamics the obvious conclusion is that it is not true.

To demonstrate it is true takes more than a claim. Currently, it just looks like confusion on the part of the author.

Perhaps the author should write a whole paper devoted to explaining how the upwards atmospheric flux can be equated with the kinetic energy – along with dealing with the inevitable consequences for current thermodynamics.

Like this:

Related

123 Responses

You have raised several interesting points of
a fundamental nature. In particular, the paper
by Toth is highly relevant. I have read it and
find it very curious with respect to how relevant
it is to Miscolczi’s derivation of a constant tau.
I just don’t know. I just don’t know if it
represents the physical situation or is just some
sort of illustration of it in one dimension.

The best I can do for now is to conclude that Miscolczi
gets to the right end point but that his paths there
are maybe not valid. I am left choosing between
rigor and intuition. My intuition is that he is, at the
end, right. There is a set point and that set point
is the optical depth. The data supports him.

For now, I go with the downward radiation being
exactly equal to the upward surface radiation.
That “exactly” greatly simplifies the anlaysis,
does it not? I find it extremely hard to make sense
of Miskolczi without that “exactly”.
m

You say that “downward radiation being exactly equal to the upward surface radiation.” No: M. says downward radiation being exactly equal to the atmospheric absorbed part of the upward surface radiation.

You lost me. M is saying that of all the photons that the atmosphere absorbs from the surface, there is exactly zero chance that the energy re-emitted will be in the form of a photon that escapes directly to space?

To calculate the empirical tau on measured profiles, eqs 1 and 2 in M2010 are needed. The result is about 1.87, both on global average TIGR and NOAA.

Is is possible to derive theoretically an equilibrium tau, simply from the four new relationships (here Aa=Ed is one of them, M7 is another). The numerical solution of the resulting set of equations is tau=1.867561…..

Let me say here again that this (M7) appeared first to him in plotting the data. This is an empirical observation. Its physical meaning in short:

Su – Fo (=Su – OLR (= G ) ) represents a net upward LW energy flow in the atmosphere, (Ed – Eu) (=G) represents a net downward LW flux (as we said: Ed is the downwelling radiative heating, Eu has it energetic source in the sum of K and F). The first is the energy into the atmosphre, the second shows how much is given back to the surface. Both of them has the same source: the incoming available energy, F°. That is, their sum equals to that.

The fits with the indiviual radiosonde measurements, look at the plots please on page 10 in the EGU presentation:

You are, of course, correct. Thank you
very much for catching me in such a
basic error and taking the time to get
back to me. The distinction that my
careless language has blurred is crucial
and central.

In talking about the climate, I find it all to
easy to form bad habits such
as using “CO2 driven global warming” when
meaning “man caused CO2 global warming”,
and, as you have just pointed out
using

“downward radiation being exactly equal to the upward surface radiation.” for ” downward radiation being exactly equal to the atmospheric absorbed part of the upward surface radiation.”

Let me say here again that this (M7) appeared first to him in plotting the data. This is an empirical observation.

[my emphasis]

So once again, an approximate empirical observation like Aa ≅ Ed gets elevated to a fundamental thermodynamic identity. If it were really fundamental thermodynamics it would be provable. Good luck with that.

Is there some prediction based on Miskolczi’s that could be tested? I’m sure there should be some more tangible consequence that would escape the blurred area of measurment approximation.
”

James Kennedy:

I believe that the data showing a decline in high altitude
humidity, quite closely matching what is
needed to compensate for increased CO2,
is a pretty good confirmation. This data is
easy to find using Google.

Are you referring to stratospheric water vapor and the Solomon 2010 publication? If so, then you might want to reread that publication as it doesn’t support your assertion:

“I believe that the data showing a decline in high altitude humidity, quite closely matching what is needed to compensate for increased CO2”

Or are you referring to Paltridge 2009? In that case it’s outcomes are disputed by e.g. Dessler and Davies 2010, Soden 2005 and (in paleoclimatology) Kohler 2010. See, for instance, figure 1 on page 2 in Dessler 2010: http://geotest.tamu.edu/userfiles/216/Dessler10.pdf

Thanks for linking the source, it appears to be in fact NOAA’s NCEP reanalysis as discussed in Paltridge 2009.

As I said in my previous post: The outcomes of Paltridge 2009 are disputed directly or indirectly by multiple studies. See for instance the issues Dessler 2010 is discussing regarding the missing tropical ENSO signal in the NCEP reanalysis.

I agree that that data I cited is disputed
in your reference. All data deserves to
be actively disputed. My mind as to high
altitude humidity data is not set.

We seem to agree that high altitude
humidity data is important.

Thanks for the citation.

I will look further into this humidity
data business to see if some resolutions of
the data disputes are emerging. I will do so because
it is my belief that the Miskolczi theory has
advanced to the point where data can now
clearly support it or refute it. If the balance
of data shows a drop of humidity corresponding
to Miskolczi’s theory, I would see that as
support of his theory. If it does not so show,
I would see that as contrary to the theory.

It’s important to understand that the computation of total optical thickness depends on total precipitable water – that is water vapor through the entire atmosphere. The location of the water vapor vertically has zero impact on tau.

However, as explained in Part One, the actual vertical distribution of water vapor has a significant effect on OLR (outgoing longwave radiation), which is why tau is not such a useful measure as it might appear.

There is a big interest in upper tropospheric humidity because of this non-linear effect on OLR. A few references were already cited.

– The paper by Toth, as well as the more general approach taken in the paper he references by Pacheco and Sañuco, show that a careful application of the Virial Theorem to the planetary-atmosphere situation results in:

(2/3) =
relating the kinetic energy to the potential energy in a homogeneous gravitational field.

(2/5) =
for a diatomic molecular gas, if you include rotational energy in the kinetic energy.

– The result attributed by Miskolczi to the Virial Theorem is:
2 =
This is inconsistent with the above results.

– In any case, the application Miskolczi makes of this equation is somewhat mysterious: relating the ratio of two energy fluxes to the ratio of the gravitational and kinetic energies, for no very clear reason.

There are no more energy balance equations we can write. There are no other constraints of energy.

You say:

Su – Fo (=Su – OLR (= G ) ) represents a net upward LW energy flow in the atmosphere, (Ed – Eu) (=G) represents a net downward LW flux (as we said: Ed is the downwelling radiative heating, Eu has it energetic source in the sum of K and F). The first is the energy into the atmosphre, the second shows how much is given back to the surface. Both of them has the same source: the incoming available energy, F°. That is, their sum equals to that.

I read words but none that I can match to any thermodynamics.

I have demonstrated that this following equation must be true for eq 7 to be correct:

2(ED – EU) = F0 [13] – to be confirmed or denied

You need to prove this equation.

Here’s my claim – and I would love you to prove me wrong on this – Miskolczi has invented his equation 7, by an imaginary balance that has no thermodynamic meaning.

What you need to do to prove it is to write down the derivation (that means equations) with reference to something like the first law of thermodynamics, or established thermodynamic relationships – which is what I did for eqs 10, 11 & 14.

Unless new information is available we can’t get any further and therefore, eq. 7 is not proven and, therefore – invented.

Which seems to mean the following equations are based on an imaginary premise.

If Miskolczi believed that this equation appeared to be true by virtue of experimental work that is what he should have written, not a pseudo-thermodynamic claim.

I feel like the alleged link between flux and kinetic energy is “way more wrong” than identifying the right ratio of atmospheric kinetic to potential energy – if you see what I mean.

Also in the 2007 paper on p4:

The atmosphere is a gravitationally bounded system and constrained by the virial theorem: the total kinetic energy of the system must be half of the total gravitational potential energy. The surface air temperature tA is linked to the total gravitational potential energy through the surface pressure and air density. The temperature, pressure, and air density obey the gas law, therefore, in terms of radiative flux:

SA =σtA4 represents also the total gravitational potential energy.

Yet, there is no direct relationship between flux and kinetic or potential energy.

Kinetic energy is linearly proportional to temperature of the gas.

So the only conclusion – accepting Miskolczi’s hypothesis – is that flux is proportional to temperature.

This goes against 100+ years of experimental work which shows that flux is proportional to the 4th power of temperature.

As Miskolczi appears to agree with the equation for flux being to the 4th power of temperature he must be confused about the relationship between kinetic energy and temperature of a gas.

Yet this is also elementary thermodynamics.

Obviously, the many people who have embraced Miskolczi’s theory have already unraveled this and so shortly someone will explain what appears to us untutored folk as “making a dog’s breakfast of thermodynamics“.

, is trivial. It certainly is even more trivial
when one is talking about sets of differential equations,
where one is not looking at absolute values but
rather entirely at rates of changes over ranges
of T less than 5 deg. c.

In the above:

F = black body radiation calculated using

F = 5.67 T^4 T in deg K

dF/dT @288 is the first differential of F with repect to T
at T = 288 deg. K
.
.

As I have just illustrated, over small temperature
ranges, it is entirely appropriate to think of
F as roughly proportional to T

When dealing with multiple interacting non-linear
equations, it is almost always necessary to
use piece-wise linearizations such as given above.

Yes, I agree: There doesn’t seem to be any reason to relate the ratio of potential to kinetic energies (whatever the value is) to energy fluxes. The advantage of focusing on the ratio was that this could, at least, be properly calculated. I was hoping to find some technical point on which to establish intellectual contact with Miskolczi.

I had a disjointed discussion with him on another site, and sent him an extensive write-up, part of which turned out to be essentially equivalent to Pacheco & Sañudo, and part of which were questions related to the first 7 or 8 equations of his paper. (Like you, I also couldn’t find any comprehensible justification for equation 7.) Unfortunately, I got no answer from Miskolczi until later; at which time I was bogged down in various job-related issues, and had no time to discuss the matter.

There was no “eventual answer”: For several weeks, Miskolczi put me off, saying he was traveling; then when he got back to me, I was up to my ears in something less interesting but more critical to my mundane existence.

Then when I had some time, he stopped responding to blog discussions in general.

Zagoni seems to be standing in for him, and I have sent him the same paper. However, he has never addressed the specific questions raised therein either.

It seems to be one of a number of conceptual points of the paper to which it would seem that there should be a crisp, clear answer. But it’s never been provided.

If you restrict movement to the vertical axis alone and ignore rotation, then 2 avg(Kz) = avg (Uz). But that’s a trivial result and doesn’t explain the teleportation (step makes it sound like you could actually get there) from energy to flux.

You can also get a simple factor of 2 if you assume a gas held together by gravitational attraction to a central mass (instead of a homogeneous g-field), but then you also get a factor of (-1), because the average potential energy is negative.

It is possible to look at each component of the total energy, three kinetic and two rotational degrees of freedom, separately as long as one remembers equipartition applies. The point I was trying to make is that the concentration on Miskolczi’s use of the Virial Theorem is a distraction from the real problems with his hypothesis.

There appear to be many problems with the structure of his argument. The fundamental problem is that one doesn’t have a clear idea of where he is starting from and where he is going. Like the fact that what seems to be presented as a theoretical argument is later interpreted as a fact derived from measurements.

In the abstract to his paper, he made a special point of highlighting the relationship that was supported (somehow) by the Virial Theorem. But when I was trying to pin down the exact relationship between what he doing and the Virial Theorem, he asked, “Why are you so interested in the Virial Theorem?” And my reason was that the Virial Theorem is a very well-defined result, which can be calculated in a number of cases; so some examination of how he was applying it could be useful as a way of unpacking his logic.

Further note: A number of folks who were convinced that there was gold in Miskolczi’s hills started a group project to re-write his papers so that the heathen could be saved. I believe they gave up after a few months, because they couldn’t straighten out his logic to get a straight-forward story: They discovered that they did not understand the paper themselves. So the missionaries ended up in the stew pot, as it were.

Zagoni still goes about various sites defending Miskolczi’s results, but the fact remains that the simplest and most effective strategy for doing that, providing a comprehensible re-write of the entirety, has never been done. Zagoni’s own power-point presentation fails to clarify the very many conceptual questions people have raised in trying to understand the original Miskolczi papers. In it, he now claims that some points that Miskolczi originally seemed to derive as theoretical results were actually extracted from observations; of course, this makes an incoherent mess out of the argumentation.

Since I find it very hard to
understand Miskolczi, but I feel he is
essentially right in his thinking that
orthodox climate science has way over-
emphasized the importance of CO2, I have
been working on my own explication of his
work. My approach has been that it is
not necessary to pin down all the details
such as exactly what the word “Virial”
means and exactly how M uses the so-called
“Virial Theorem” before one looks at what
the data says and then tries to see that data
as compatible with the overall climate picture
M presents.

My starting point in explicating M’s theory
is to come up with a reasonably accurate
but much simpler derivation of “tau”.
I could go into that in detail if you would like.

I would be very interested in seeing an intelligible discussion of Miskolczi’s theory. But from my point of view, it has to be intellectually coherent: Not a strange melange of “data” and “concept”, but something with a clean set of ideas, assumptions and logical/mathematical reasoning; supported by observational data when the role of these data has been made clear.

You could start with a definition of M’s “tau”. To be honest, I have never gotten far enough in his paper to see what its role was (I only believe in crossing two or three incomprehensible rivers at a leap). So please start from the beginning.

The paper was written under the auspices of NASA and used NASA data; on page 33 M concluded a feedback climate sensitivity for 2XCO2 = 0.48K. This was based on a derived Tau using all greenhouse molecules and covering all the regions of the Earth. The fact that this paper and M’s subsequent papers were clear-sky derivations seems to be a sticking point; I don’t know why; isn’t the clear-sky the callibration point; and as Steve Short has calculated above the clouds are clear skys.

M’s subsequent papers have a climate sensitivity of 0, reflecting MEP. Some comments above cast aspersions on whether there is any concrete evidence for M’s theory, especially in respect of water vapor levels; in particular Paltridge’s work is denigrated on the basis of Dessler’s papers; pardon me if I don’t prefer Paltridge’s work:

b) the usefulness of total transmissivity (=e-tau) given that it can remain constant while OLR can change

c) the values of the various “greenhouse” gases over the 60 years used to compute the average global tau

Another question – not raised before – that I am unclear about is how global tau is computed. It may well be in one of the papers. I can see how tau is calculated and of course it is the right equation. But how is the global value calculated? Area weighted? Weighted by OLR?

On a couple of interesting questions you raise I will pick those up in another comment later.

The fact that this paper and M’s subsequent papers were clear-sky derivations seems to be a sticking point; I don’t know why; isn’t the clear-sky the callibration point; and as Steve Short has calculated above the clouds are clear skys.

Can you elaborate?

Many papers differentiate between clear skies and cloudy skies because the measurements of various parameters like water vapor are not well-known under clouds. So with much higher confidence for non-cloudy skies, it is a more verifiable “calibration point”.

But if you want to compute the total optical thickness and you leave out 60% of the sky which has the highest optical thickness, won’t you expect skewed and probably useless results?

Let’s put it like this – if you average up the global rainfall and leave out the results under cloudy skies, will anyone take you seriously?

In respect of Aa=Ed which are quantified on page 24 of the 2004 paper, which is where we should stick to, Ed is calculated on page 21-23; Aa is derived from the Su amount and the OLR, both quantified. On page 25 onwards M derives the globally averaged Tau; on this basis I don’t see how Tau and OLR can be distinguished; Tau is variable but the average amount is constant; to me this suggests the same problem that occurs with comparisons between GAT and OLR where unless SB is powered at every site before being averaged instead of being powered and the averaged there will always be a difference between an averaged measure of energy balance be it temperature or optical depth and OLR.

I started to think about all this SoD and had a dreadful feeling of deja vu; this discussion on the Tau and eqn 7 and Virial and Kirchoff will explain why:

Thanks for posting the write-up I sent to Miskolczi, as I was planning to do; and by the way, it’s not by “Ian Neale”, it’s by me, Neal J. King: as indicated in the field designated “From” !

Further to that point: Questions are raised concerning equations up to #7 in the Miskolczi paper.

All of this was based on my attempt to understand the two papers mentioned heretofore. So, as far as I am concerned, you haven’t provided any new information or perspective. In other words, we’ve been down this road before, and nobody has ever arrived at the goal: a convincing explication of what M was getting at.

As stated earlier, the only thing that is going to convince people as a group is an explanation of the whole theory that is clear and consistent, from the beginning.

Yes, sorry Neal, I looked at that thread on Niche and became dizzy so I misspelt your name; SoD’s point about a constant OD and a changing OLR is a new slant but a cul de sac I feel since M makes it plain in the 2004 paper, the significance of which everyone seems to be ignoring, that OD is calculated using OLR:

“The theoretical and simulated global average graybody optical thicknesses are in pretty good agreement: they are 1.87 and 1.86, and they correspond to a vertical optical thickness of about τ =1.23 . These values are also the indication that the clear atmosphere loses its thermal energy close to the peak efficiency, see Fig. 15. Using Eq. (1) with the same data would result in about 9% higher
optical thickness, and using Eq. (2) would result in an unrealistic low value of 0.67. Just for reference, if we use a global average surface and surface air temperature of 288 K and the ERBE global average clear-sky OLR of 268.0 W
m–2, the theoretically estimated vertical optical thickness would be 1.17, which is reasonably close to our global clear-sky average.”

The problem is that we don’t actually have much data on OLR or albedo for that matter. We certainly don’t have 60 years data. The first ERBE (Earth Radiation Budget Experiment) satellite was launched in 1984. Not only that, but the data isn’t very precise. Any claim that τ has been nearly constant for 60 years is based solely on calculated values and a lot of assumptions.

I believe there is very little that is clear in Miskolczi’s papers, which is why the discussion has never come to a satisfactory conclusion.

If you believe the theory is well-enough advanced to be compared sensibly with observations, you should have a sufficient understanding to be able to present it. Otherwise, you’re just selling odds on a horse you’ve never seen.

I’m by no means an expert on M’s theory and haven’t investigated much on this point, but I’ve always assumed that the ‘Virial Theorem’ aspect of M’s assumption was to do with the ‘Clausius Clapyron’ (CC) relationship that ‘switches’ atmospheric energy between a kinetic value and a potential value (latency) by instigating a phase change of H2O.

This ‘switch’ has a very small hysteresis value and is constantly changing for different atmospheric regions and altitudes, but its averaged value at near surface altitude seems quite static within defined regions (perhaps because ‘RH’ (relative humidity) regulates the evaporation of ocean surface and land surface water). When coupled with land and ocean surface insolation quanta, the CC relationship may also predict the phase changes between ‘La Niña’ and ‘El Niño. However, I’m no expert.

A good give-away is your conflation of the vapor/liquid phase transition with the La-Niña/El-Niño phase transition. This is slightly less plausible than producing raisins by drying grapefruit, or by adding water to grapeshot.

“When coupled with land and ocean surface insolation quanta, the CC relationship may also predict the phase changes between ‘La Niña’ and ‘El Niño.”

Whilst bearing in mind that ~70% of global insolation strikes ocean surface and ocean ‘thermal capacity’ (Cp) is greater than land Cp, ocean Cp can be seen to provide energy to land Cp where land surface temperatures are low (as can be seen from diurnal atmospheric observations between land and ocean surfaces). I see no reason why this effect shouldn’t also hold for larger, global, events.

Also, it’s hard to understand how the ‘Viktor T. Toth’ paper is pertinent to M’s theory! From the abstract:

“We derive a version of the virial theorem that is applicable to diatomic planetary atmospheres that are in approximate thermal equilibrium at moderate temperatures and pressures and are sufficiently thin such that the gravitational acceleration can be considered constant. We contrast a pedagogically inclined theoretical presentation with the actual measured properties of air.”

Ocean surface ‘evaporates’ generating H2O ‘water vapour’ (WV). WV is a ‘triatomic’ gas and the paper was written for “diatomic planetary atmospheres”. Moreover, WV changes phase within the atmosphere, by nature, in Earth’s atmosphere.

Yes, I’ve seen that link before, it hasn’t changed much. There are various forms of calibration that relate to the ‘saturation point’. From the ‘contained mass’, where pressure and temperature change within the fixed volume of a ‘pressure cooker’, to the ‘free’ atmospheric ‘parcel’ that is lossy for mass! We need to be careful when selecting the means of calibration, or we may ‘throw the baby out with the bath water’.

“It has exactly nothing to do with quanta or weather phases.”

Then why, when insolation quanta are low, does a high humidity atmosphere generate cloud (or mist) with lowering temperatures?

..SoD’s point about a constant OD and a changing OLR is a new slant but a cul de sac I feel since M makes it plain in the 2004 paper, the significance of which everyone seems to be ignoring, that OD is calculated using OLR:

Not according to the formula in the paper.

Optical thickness cannot be calculated using OLR. Well, it can be empirically determined from measuring OLR and knowing upward surface radiation. But this is not the method in the paper.

M2004 says: “Regarding the zonal temperature change estimates (in Table 6), once again we have to
emphasize that keeping the zonal OLR as a constant (while changing the optical thickness) is not realistic. There exists only one overall constraint, and that is for the global average OLR.” [page 33]

This is shown in EQN 9 and depicted in Figure 24; Figure 24 is plotted from Table 5 ERBE data points.

I know aspersion have been made about the ERBE data, and the Radiosonde data, but that is what we have and can’t be blamed for relying on it.

Ok, so work backwards; as M 2010 says: “Except for OLR, ED, and SU, the above radiative quantities cannot be directly measured.” OLR, ED and SU are derived in EQNs 3&4 from 1&2 and verified, if you believe/accept the ERBE and Radiosonde data, in EQNs 8&9.

I don’t know, I might be missing something but it seems to me that OLR and OD are connected by real time point data; the intervening non-empirical terms are deduced from the 3 measured variables and the produced EQNs cross-referenced against various known variables such as CO2, H2O and temperature.

I read somewhere that there is concern about M2010’s statistical testing but I forget what the complaint was.

Ok, eqns 1&2 show OD is established by point data averaged globally; is that right?

Eqns 3&4 convert the empirical and computed values from eqns 1&2 into the M forms from which OLR = EU + ST is derived; since OLR is known independently of that process and gives EU and ST, the known value of OLR can be used to check the validity of eqns 1&2 by working backwards.

The complaint about statistical testing was that effectively there wasn’t any. He claims that the slope of τ over time was not significantly different from zero, which is likely true considering the noise in the data. But then he asserts with little evidence that if τ followed standard theory that the slope would be different from zero or different from his calculated slope. It wasn’t at all clear, like a lot of the paper. It’s also not at all clear, considering that satellite data on OLR didn’t start until 1984, how the τ time series was obtained. If it was calculated, as I suspect, then his conclusion of stability appears to rest on circular logic.

From what I’ve seen in the way of contributions to this thread over the last few days, nobody here understands the Miskolczi papers.

I wouldn’t go that far. I think that we have established beyond a reasonable doubt that the papers don’t prove the basic claim that τ must be a constant according to fundamental thermodynamics. No such proof is presented in the papers. The link of surface emission flux to the kinetic energy of the atmosphere only is by assertion and fails on close examination. The claim that Aa ≡ Ed is also by assertion and fails close examination (except perhaps for low cloud covered sky, which is particularly ironic). The validity of Equation 7 in M07 depends on conditions that don’t exist in the real world, a 1D, single slab, cloud free atmosphere, constant illumination and a surface heat capacity of zero.

M10 does not support M07 because the data used to calculate τ is cloud free. The claim that clouds don’t make a difference is unsupported.

Before these posts, I couldn’t point anyone to clear, concise analysis of Miskolczi’s errors. Now I can.

Is this a test Neal? After all, what else can you do with things you don’t understand except believe, or not, as the case may be.

Or, plug away and hope for some enlightenment; in discussion with someone once, Stokesy, I think, I went away and learnt how to do partial differential eqns; what a waste of a week; came back and argued with Stokesy who bamboozled me further, then someone else pops in and wipes the floor with Stokesy. It all evens out in the long run.

If I don’t understand something, I don’t understand it. It’s not necessary to assume either that it is right or that it is wrong.

Of course, when one sees inconsistencies and logical failings, one starts to develop a strong feeling that it doesn’t make sense. I never give anyone the benefit of the doubt unless I know them and their work: If something doesn’t make sense, I’ll push on it.

..Also, it’s hard to understand how the ‘Viktor T. Toth’ paper is pertinent to M’s theory..

..Ocean surface ‘evaporates’ generating H2O ‘water vapour’ (WV). WV is a ‘triatomic’ gas and the paper was written for “diatomic planetary atmospheres”. Moreover, WV changes phase within the atmosphere, by nature, in Earth’s atmosphere.

Word matching is completely different from understanding science.

Here is an extract from Toth’s paper:

Air, in particular, is composed mainly of diatomic gases (notably N2 and O2), which at room temperature have two rotational degrees of freedom in addition to the three translational degrees of freedom discussed above. (At higher temperatures, vibrational modes also play a role.) The kinetic energy of a column of gas is its internal thermal energy.

If you want to know the energy in the atmosphere then you need to know the energy of the major constituents. N2 and O2 are 99% of the atmosphere and are diatomic. Total energy = energy of N2/O2 + energy of trace gases. You can do a more accurate calculation using the fractions of each gas but it won’t change much.

And note that I haven’t made a point out of the use or abuse of the virial theorem – there are bigger problems with the paper before we get to this one.

Thank you for your polite reply SoD. There are a couple of points that arise from it that I would like to address.

“Word matching is completely different from understanding science.”

As an engineer I can understand this POV, however! 🙂 My “Word matching” throws up the issue that the paper is only useful for ‘clear sky’ applications, when M’s calculation is based on ‘real sky’ radiosonde observations and ‘real sky’ observations from a high tower. Is this not so (anyway, from what I’ve read, Miskolczi leaves the ‘theorising’ for the ‘apparent discrepancy’ to ‘others’ for discussion)?

“You can do a more accurate calculation using the fractions of each gas but it won’t change much.”

How so? Water vapour is a ‘trace fraction’ of Earth’s atmosphere, but is recognised as the ‘major transporter’ of energy from the surface to the tropopause. That is, unless you’re ignoring ‘latent transport’ and only tackling the ‘radiative’ aspect of Miskolczi’s paper (which, BTW, may throw up the hysteresis value for ‘phase changes of water vapour’). How do you differentiate these ‘modes of energy transport’ within the troposphere?

As I don’t have much background in hardcore atmospheric measurements, I have the most interest in getting a clear conceptual and theoretical picture of what is supposed to be going on. I think there has been quite a bit of confusion between what is assumed in theory and what is observed in measurements. Indeed, until the theoretical framework is properly defined, it seems unclear to me what it is that can be said to be observed & measured. What does one mean by measuring tau if one doesn’t know what tau is or is supposed to be?

That’s not a particularly good sign: Right after Einstein’s final formulation of General Relativity came out, there were said to be 3 people in the world who understood it. And Feynman thought that was likely a gross underestimate: He figured that once it was published, there were lots of people who understood it.

You did not show, that the atmospheric Kirchhoff law is not valid, you did not show that the virial concept is not applicable for the atmosphere, you did not show that the Su=OLR/f radiative equilibrium rule is not valid, you did not show that NOAA R1 dataset is wrong, you did not show that the TIGR 2 dataset is wrong, you did not show that Su-OLR is not equal to Ed-Eu, you did not show that (Su-OLR)/Su is not 1/3, and you did not show a single article (peer reviewed) where the global average tau is different from 1.87.

The idea that τ must be constant comes from the solution of a simple model that bears little relationship to the real atmosphere, the bounded semi-transparent gray atmosphere with constant surface energy flux. The real atmosphere isn’t gray and it has clouds, i.e. it isn’t semi-transparent everywhere. What I also don’t see in the derivation is the atmospheric temperature profile.

I’ve looked at van Andel’s paper in E&E and am not impressed. Example:

It leads to a 25 W/m2 unacceptable underestimation of the window radiation.

(p.291).

No, the failure to consider that clouds are opaque to IR leads to a 25 W/m2 overestimation of window radiation. Clouds don’t have a window.

For that matter, even when there are no clouds, the atmosphere is effectively opaque in certain wavelength regions like the center of the CO2 band at 667 cm-1.

The problem with a gray atmosphere is that it redistributes the spectral energy absorbed over the full frequency range at each altitude. As a result, a gray atmosphere has a greenhouse effect even if it’s isothermal. The real atmosphere doesn’t. Of course neither the real nor a gray atmosphere can stay isothermal, as the top will always radiate more than it absorbs and so cool until a stable lapse rate is established.

I have been working on the Virial Theorem stuff, because it’s conceptually straightforward and gives computationally definite results. The coupling into the rest of the paper is the ratio of the potential to kinetic energies, which is somehow supposed to equal a ratio of fluxes. Without getting into the rationale for this, at the moment, can anyone tell me what happens to the rest of the theory if this ratio changes from Miskolczi’s value of 2 to the value 2/3? Does it change anything?

“can anyone tell me what happens to the rest of the theory if this ratio changes from Miskolczi’s value of 2 to the value 2/3? Does it change anything?”

It changes a lot. It would mean to me that Eu can not be equal to Su/2 and the mystery tau can not be 1.87. See slide 19 in EGU 2011-13622. The Su=2Eu dot must lie on the theoretical line. You stopped reading my paper at page 10 so you do not know the rest of the theory. Why do not you read further? The paper is about the Su=OLR/f relationship and for this you do not need the Kirchhoff or the Virial rule.

The Virial theory and its application is not as straightforward as you think. See Cox and Giuli, 1968, Principles of Stellar Structures , page 408 :

“The virial theorem may be expressed in a variety of different forms
and also may be interpreted in a number of different ways. It should
be pointed out that the virial theorem need not necessarily apply to
the entire system, but may apply to only a part of the system.”

The problem with the Kirchhoff law is the same. It has too many faces, and in fact there is a long way to go from the directional Kirchhoff law to the emissivity. Plenty of room for SoD to say that Miskolczi is wrong, because Miskolczi computation is wrong. I should say that SoD computation is wrong and apparently we are talking about different things, which renders the whole discussion about this issue useless.

I applied the virial concept for a hydrostatic system and considering the vertical transfer of momentum only, I found that P/T=2. (BTW, Viktor Toth agreed with this, but he did not make his changing opinion public, and also Jan Pompe come up with the same result). If you recall the Vogt-Russel theorem you may recognize that the P/T=Su/Eu relationship is not that absurd. The ad-hoc feedback parameters in GCMs are far more idiotic than this.

The fact that you, Toth, Levenson, Sanudo&Pacheco etc. have different ratios for P/T shows that we are not interpreting the virial concept in the same way. Years back when I wrote the paper I computed this and I still insist, that my computation is correct, and the Su/Eu ratio may very well be constrained by the virial theorem.

I hope you will not take this as ‘word salad’.
And, I think suricat has interesting and important message that you should not ignore.

– I did not read further than about equation 7 of the 2006 paper because I don’t like to proceed too many steps beyond where I am sure of what the issue is. When the conceptual issues are not clear to me, I prefer to get clarification.

– The Virial Theorem is something that is pretty well-defined and comprehensible: It relies on the system being bounded in space, but is otherwise not very tricky. Particular examples can be calculated explicitly, allowing a good check on one’s understanding. However, I do not believe the actual meaning of the Virial Theorem can be applied to one dimension. The reference to Cox and Giuli does me no good, since I am in Munich and English-language textbooks are infrequently available. The reference that I have relied upon is: George W. Collins, II: The Virial Theorem in Stellar Astrophysics, http://ads.harvard.edu/books/1978vtsa.book/ , which is easily accessible and, I believe, fairly complete. The point is that the time-averages are related as:
2[KE] = -[r . F]
for spatially bounded systems: I do not find the concept of “vertical” kinetic energy to be a useful or coherent concept. The general definition of KE is mv^2/2, and it is a scalar quantity. Toth’s article does not distinguish KE from internal energy (including rotational energy), which I think is confused. In terms of the usual definition of KE (for which the average KE of a molecule = (3/2)kT), the straightforward result of applying the Virial Theorem to a homogeneous gravitational field is:
(2/3)[KE] = [PE],
as I have demonstrated previously (and also I have updated the text to recover a Word version, in the last few days). I will send the new version to you.

I regret to state that the fact that Toth and Jan Pompe affirm your result is, for me, not much of a recommendation. Toth’s presentation of the Virial Theorem is insufficiently general and lacks perspective; and in my experience, Jan Pompe cannot write three consecutive equations without erring in one of them. I would prefer to see your own calculation; or you can comment on mine.

With regard to the final results, there is not a significant difference between mine and Pacheco & Sañudo: we agree on essentially everything, although our methods differ. I do not know Levenson’s paper(s). Between mine and Toth, the issue is that he is confusing or conflating the terms “internal energy” and “kinetic energy”. The Virial Theorem applies to translational kinetic energy (in 3 dimensions). By virtue of the equipartition theorem of statistical mechanics, it’s easy to calculate the internal energy due to rotational or vibrational modes as well, if necessary; and this is what Toth does. However, strictly speaking, this is a separate issue from the Virial Theorem. When we are talking about the same things, we agree.

With regard to the discussion concerning Kirchhoff’s law, the issue seems convoluted to me, so I have put it aside.

With regard to suricat’s ideas: When someone conflates the ideas of liquid/vapor phase transition with an El-Niño/La-Niña phase transition, and tries to bring in the Clausius-Clapeyron equation, I have to assume they are arguing by free-association. Hence my reference to the Turing test, which measures the success of a computer program’s ability to model consciousness by the tester’s inability to distinguish it from a human being, during keyboard interaction.

I will send you the updated Virial Theorem discussion.

Regards,
Neal

Beyond the formal derivation using the Virial Theorem, the factor of (2/3) can be proven by direct calculation of [KE] and [PE] for simple models such as the isothermal perfect gas or, with more difficulty, the adiabatic perfect gas.

Thank you for your polite reply SoD. There are a couple of points that arise from it that I would like to address.

And thank you for your sense of humor.

If people have questions I try and answer them.

When people dismiss carefully worked-out papers because they don’t appear to have made any attempt to understand them, I tend to put more effort into caustic comments. It’s a character flaw.

How so? Water vapour is a ‘trace fraction’ of Earth’s atmosphere, but is recognised as the ‘major transporter’ of energy from the surface to the tropopause. That is, unless you’re ignoring ‘latent transport’ and only tackling the ‘radiative’ aspect of Miskolczi’s paper (which, BTW, may throw up the hysteresis value for ‘phase changes of water vapour’). How do you differentiate these ‘modes of energy transport’ within the troposphere?

What has the the efficiency of latent heat transport by water vapor got to do with calculating the total potential energy or the total kinetic energy in the atmosphere?

Water vapor is massively more effective than N2 or O2 in absorbing radiation in the troposphere. But this is also irrelevant for calculation of potential energy and kinetic energy.

Have a read of the paper by Toth and try and follow the equations. He has put some work into explaining the background in easy to follow steps.

At each step ask yourself if the equation is correct and what effect mechanisms like heat transport have in each equation.

I don’t think I can explain it any clearer than Toth and if you don’t take the time to work through that paper, not much point me explaining the same points.

If you have tried to understand his paper and can’t figure out something then, of course, you are welcome to ask.

Sorry for the delay. I’m my mother’s main carer, she’s 99 this month and uses most of my time 24-7. However, I found the time to read Toth’s paper again and still don’t see ‘Cp’, or ‘latency’ mentioned anywhere. You could try to point to it for me, but as it’s not there I don’t see how you could do this.

Eq. 26 introduces Cv, but this is a ‘specific heat’ term for a ‘molecular gas’ and nothing to do with atmospheric water and ice. Thus, I can only reiterate that the paper is for use with a ‘clear sky’ model. We need more math to introduce a ‘real sky’ model that can expand on Ferenc’s observations.

Where do we go from here? How about I answer one of your questions and you answer one of mine (to the best of both our knowledge). Although I asked first, I don’t mind being the first to answer.

“What has the efficiency of latent heat transport by water vapour got to do with calculating the total potential energy or the total kinetic energy in the atmosphere?”

Thermodynamics operates at a 100% efficiency rate. It’s not about efficiency, it’s about ‘thermal capacity’ (Cp).

To use an analogy, the atmospheric side of the hydrosphere behaves like a ‘rechargeable battery’ that begins its life at the surface as a ‘fully-charged molecule’ (water vapour) that’s lighter than air. Due to ‘climate cell convection’, and the property of being lighter than air, a ‘water vapour’ (WV) molecule finds itself rising to increasing altitudes through the ‘adiabat’ (as altitude increases the temperature of an ‘adiabatic atmosphere’ decreases) of the troposphere.

Eventually, the lower thermal kinetic of a reduced temperature at altitude causes our WV molecule to ’emit radiant energy’ (latent heat of vaporisation) as it ‘changes phase’ to water. This usually happens when the molecule encounters a comparatively large mass that absorbs its kinetic, such as a small water droplet, soot particle, or spec of floating dust (I’ll include ‘cosmic particulate ‘radiation” here) at this lower temperature. This is the first stage of our ‘rechargeable battery’s’ energy discharge and it’s now well ‘within’ the atmosphere.

From this point our WV molecule can, by hapstance, either re-evaporate with a recharge of thermal energy, or discharge (with greater temperature reduction and another loss of ‘radiant energy’ during ‘change of phase’) further to ice.

Water is heavy in the atmosphere, as is ice, and precipitation may be the order of ‘hapstance’, but I’ll not speculate on the outcome of a supposition as I think you should already grasp the importance of Cp from this post. It ‘buffers’ a ‘radiative component’!!!

Thermodynamics has a lot of partial differential equations where you hold one thing everything constant except the variable of interest. One can hold pressure constant and vary temperature and volume or hold volume constant and vary temperature and pressure. Cp is the heat capacity at constant pressure, Cv is the heat capacity at constant volume. For liquid water, they’re approximately equal. For a gas, not so much. Physicists like to do thermodynamic calculations at constant volume as the equations are simpler, or that’s what I remember Feynman saying in freshman physics. Chemists use constant pressure because that’s how most chemistry experiments are actually done.

Thanx for that DeWitt. My ‘bruv’ was an analytical chemist (RIP) and I did learn a bit from him. However, as an engineer, it’s ‘normal practise’ for me to use Cp as well.

The problem with the use of Cv (fixed volume) with an atmosphere bounded only by gravity is that the original volume changes. It has to, as there is no fixed boundary to contain the original contents of the ‘parcel’.

I just find that because there are no fixed boundaries for an atmosphere, Cp (fixed pressure [properly written as Cp, e]) is the more appropriate term (though weather forecasters would disagree).

Ideally, we should observe Cm, e. However, gasses diffuse and mass or energy is also lost or exchanged from the original sample (heck, when you use a ‘bomb calorimeter’ the energy used to raise the fuel and oxygen to a combustion level and the energy introduced by the ignition ‘spark’ are accounted for, but there are still ‘other’ losses).

So it’s ‘swings and roundabouts’ really. We can’t observe all the original bits after they’ve changed so we run the whole range of C# until we have the gist of interactions from the viewpoint of all stationary parameters as a theoretical model. This is why I choose Cp, e to represent an open system. The volumetric change for the evolution of water vapour, with high energy input and minimal temperature change, is enormous.

..However, I found the time to read Toth’s paper again and still don’t see ‘Cp’, or ‘latency’ mentioned anywhere. You could try to point to it for me, but as it’s not there I don’t see how you could do this..

I don’t know where to start. You seem so sure that you are on the right track.

4. Kinetic energy of a gas = Nm/2 = 3KT/2, where N is the number of molecules and the other terms are defined under the heading “Kinetic Energy of a Gas” in the article.
And if different regions are at different temperatures we can simple divide up the regions and apply the above formula and sum the results.

5. Principle of equipartition of energy – (e.g. here), where internal energy of a gas (or liquid or solid) is equally divided among all the degrees of freedom (so long as LTE applies).

Let me know if we can tick these off as common ground.

And for newcomers to statistical thermodynamics, the principle of equipartition of energy allows successful prediction of properties like the specific heat capacity of a gas.

Again! Yes (for the most part), but no! KEH2O has another two degrees of freedom. Ice and water.

“4. Kinetic energy of a gas = Nm/2 = 3KT/2, where N is the number of molecules and the other terms are defined under the heading “Kinetic Energy of a Gas” in the article.
And if different regions are at different temperatures we can simple divide up the regions and apply the above formula and sum the results.”

For the case of Earth’s atmospheric condition from ‘that paper’, absolutely not. The paper doesn’t address the full range of ‘freedoms’ enjoyed by H2O. Thus, the theorem is unable to address these ‘freedoms’ for an accurate ‘summation’ and final analysis

“. 5. Principle of equipartition of energy – (e.g. here), where internal energy of a gas (or liquid or solid) is equally divided among all the degrees of freedom (so long as LTE applies).”

To a point I concur, but Earth’s systems are in continual flux and alter the boundary demarcation points between ‘phase changes’. Thus I ask, how can ‘LTE’ (local thermodynamic equilibrium) exist anywhere within an ‘Earth-like’ system? Though I’d like a ‘closure’ for this and this involves Cp/latency. IMHO LTE in the troposphere is a fallacy and the thermal capacity of Earth’s atmosphere at constant pressure only ‘appears’ ~constant because of latent interaction by H2O’s phase changes. This comes back to the ‘real sky/clear sky’ argument again.

Tom Vonk didn’t convince me of his understanding of LTE before Steve closed the Climate Audit Forum, but you have a chance to influence me now.

I’m being ‘open’ with my ‘previous’ and I hope you’ll display the same candour.

You did not show, that the atmospheric Kirchhoff law is not valid, you did not show that the virial concept is not applicable for the atmosphere, you did not show that the Su=OLR/f radiative equilibrium rule is not valid, you did not show that NOAA R1 dataset is wrong, you did not show that the TIGR 2 dataset is wrong, you did not show that Su-OLR is not equal to Ed-Eu, you did not show that (Su-OLR)/Su is not 1/3, and you did not show a single article (peer reviewed) where the global average tau is different from 1.87.

So what is flawed?

That’s a fascinating approach you have to theory.

I asked you to prove two equations which are apparently “made up” and you don’t feel you need to prove them.

They appear out of thin air and I have proved one of them wrong.

Your response is to claim that other stuff is correct (as yet neither accepted nor denied by me).

This does answer some questions about your theory and about your approach to the idea of theory.

My approach is old-fashioned. I believe that if, with development of a theory, equation 3 builds on equation 2 (these numbers are just as example) and equation 2 is false, then even if equation 1 and equations 3-100 are correct – the theory is still false.

Unless the writer of the theory can redevelop the argument without flawed equation 2.

Equation M7 is simply wrong. To prove that it’s wrong, assume a transparent atmosphere where Ed = Eu = 0. Plug that into M7 and you get Su – Fo = Fo or Su = 2 Fo. But we know that in a transparent atmosphere Su = Fo. M7 can only be correct if Fo = 0, a rather trivial case.

By varying the water vapor and CO2 content of the atmosphere using MODTRAN, adjusting the surface temperature offset to keep OLR constant at 100 km, it’s clear that Ed-Eu and Su-OLR aren’t constant as tau changes, as should be expected. Both increase as tau increases.

I am going to give up with you again. I do not have time for this. I do not care how you play with your MODTRAN toy, and what kind of irrealistic atmospheric structures you create. Be happy with them.

But think a bit about what you wrote: You increase the Su while keep
the OLR=St+Eu constant. How can you do that? How can the MODTRAN do that? Or how a real atmosphere can do that? How can you vary the H2O content? H2O is depending on the thermal structure I guess. How did you alter that? What kind of global average atmosphere did you use?

OLR must be very close to equal to Fo or there will be a substantial radiative imbalance which will lead to heat loss or gain depending on the sign of the imbalance. When τ is reduced, St and OLR increase if nothing else changes because Eu is constant. But of course, something does change: the surface temperature. A reduction in surface temperature lowers Su and cools the atmosphere so Eu is lower too. Reduce the temperature enough and OLR = Fo again.

This is all really basic stuff. I’m surprised you don’t understand. MODTRAN is crude, but it gets numbers in the ballpark as demonstrated by the relatively good agreement between your calculations for US76 at 288.2 K surface temperature and mine using MODTRAN.

So for what range of τ is equation (7) valid?

That’s really a moot point, though, as your calculation of the optimum B0 is wrong. When the fact that Bg is a function of τ is included, B0 becomes a monotonic function of τ.
B0 → OLR/2 as τ → ∞
B0 → OLR as τ → 0

You should notice, that Eq 7. was not meant to be valid when there are no GHGs in the atmosphere. For that – if you read a bit further – there is the Su+St/2-Ed/10=3OLR/2. (This is my other invention, it looks you and Neal have problems with deriving it .)

Eq. 7 simply states, that the sum of the Su-OLR and Ed-Eu terms – in ideal greenhause case – must be equal to Fo. I assume that the complex dynamics of the system may support this assumption, and will explain the Su=3OLR/2 (global average) observed relationship. In case of the Mars Su-OLR+Ed-Eu=Fo-St, that is, no clouds, no ocean currents, no general circulation, and most inportant, no SW absorption in the atmosphere. Perhaps you may have other explanations….

Eq. 7 simply states, that the sum of the Su-OLR and Ed-Eu terms – in ideal greenhause case – must be equal to Fo. I assume that the complex dynamics of the system may support this assumption, and will explain the Su=3OLR/2 (global average) observed relationship.

I am glad that we seem to be making progress in clarifying what has, and what has not, been shown. When ALL these issues are worked out, I would look forward to reading a clear re-write that starts from the beginning and sets out the assumptions and derivations straight-forwardly.

But I see that there are still many open issues. SoD has reminded us of the Eu/KE question (Section 3.1), which is how the Virial-Theorem discussion gets coupled into this paper. I will be very interested in that topic.

Further, I give warning that I have discussed the maximum-entropy principle application to this paper a few years ago, and as far as I am concerned, the case remains quite difficult to understand.

However, I don’t wish to focus on the MEP now, as we are still coming to terms on the Virial-Theorem discussion.

(For those who may be curious about the status of this discussion: I have sent Ferenc an updated, more explicit, version of my calculations, and he is still reading it. He has raised some questions concerning the derivation, which reminded me of some of the odd points that I had forgotten about; so I have provided some clarification; and at some point will revise my calculation again, to provide a better introduction. So I would say we are still at an early stage of discussion.)

My papers were reduced to about 40 pages by the editors due to page limitations. I did not encrypt anything intentionally. Apparently too much was assumed from a casual reader, and I left out important details. But your attitude – what you do not understand is wrong – will take us nowhere and certainly will not contribute to the understanding of the greenhouse effect.

The virial concept has a special form which tells that half of the gravitational potential energy must be lost by radiation. Further on there is the well known mass-luminosity relation for stars. The details
how P/T=Su/Eu comes along will be discussed in details later. (And again, here we are talkig about global average hydrostatic atmosphere.)

Meanwhile you may prove that P/T is not equal to Su/Eu. It seems that this rests on your belief. Now I am learning what Neal computed, and hopefully we shall reach some conclusions upon which we shall agree. Appartently the key issue here is the degrees of freedom of the considered system.

The greenhouse effect is a complex problem and there are still plenty of things to explain. But you are nowhere close to see those problems, and
with your modified or artificial atmospheres and MODTRAN runs you will never see them.

you ask:
“So for what range of τ is equation (7) valid?”

Do not you see that Eq. 7 is for global average? The Su=3OLR/2 is for global average.Instantaneous OLR/Su ratios vary with the instantaneous h2o.

Focus on the tau=1.43 that we agreed upon. How did you or your MODTRAN computed that? Why do not you compute the tau for the NOAA R1 annual mean profiles?

I think it is best if I write an article explaining this subject more fully.

Internal energy is a different property from kinetic energy. This is why the specific heat capacity depends on the degrees of freedom of a material. Because some energy goes into kinetic energy – and therefore temperature – and some goes into rotational and vibrational energy.

Kinetic energy = mv2/2. This is absolute basics and hard to know how to convince you if you think it is something else.

Degrees of freedom is different from phases (gases, liquids, solids). Degrees of freedom is a concept from statistical thermodynamics.

LTE is a well-defined state that is nothing to do with being in Thermal Equilibrium (TE). You can read Max Planck’s definition of TE in Note 2 of Part Two. And for LTE see Planck, Stefan-Boltzmann, Kirchhoff and LTE. I read Tom Vonk writing about LTE on Wattsupwiththat and he appears to be at odds with all of the textbooks so unlikely to being close to correct. Although his article was so confusing I couldn’t be certain what he actually thought.

If I lay out the basics more thoroughly in a new article perhaps the subject will become clearer.

My papers were reduced to about 40 pages by the editors due to page limitations. I did not encrypt anything intentionally. Apparently too much was assumed from a casual reader, and I left out important details. But your attitude – what you do not understand is wrong – will take us nowhere and certainly will not contribute to the understanding of the greenhouse effect.

I am happy to review new information. I can only work on what is available.

Your supporters, despite their enthusiasm for your work have also contributed to the confusion.

I hope you can take my criticism in a constructive way – if you develop what is claimed to be theoretical proof that actually is experimental proof for experimental results the readers have every right to point out that it is not theoretical proof.

40 pages gives a bit of room to explain this and I believe – just my opinion – this is actually the most important and fundamental point to explain in your thesis.

Leaving out the basis of your theory (“they are actually my results”) is a terrible oversight. You tossed out the wrong bit in your 40-page “precis”.

If you don’t understand this point, I can only shake my head.

– Perhaps you are so entranced with your new idea that you haven’t realized the rest of the world will be skeptical.
– Perhaps you don’t understand what is theory and what is experiment.
– Perhaps you took a wrong turn and would like to put it right.

My unimportant advice – if you can’t explain your results in terms of already proven theory, be upfront about it.

..The virial concept has a special form which tells that half of the gravitational potential energy must be lost by radiation. Further on there is the well known mass-luminosity relation for stars. The details how P/T=Su/Eu comes along will be discussed in details later. (And again, here we are talkig about global average hydrostatic atmosphere.)..

If your special form of the virial theorem has radiant flux = kinetic energy then:

1. The virial theorem is wrong, or
2. The virial theorem has been misapplied, or
3. The basic equation relating kinetic energy = temperature (mv2/2 = 3KT/2) is wrong, or
4. Blackbody flux is not proportional to T4

The proof of item 3 is quite basic and in all the textbooks so it seems unlikely you can overturn this one.

The proof of item 4 is based on well over 100 years of experimental work so also unlikely, and can be derived from Planck’s law which also has a solid 100 years behind it.

What is the relationship between item 4 and a real non-blackbody, non-gray atmosphere?

The graph below compares blackbody flux to the flux from a non-gray atmosphere, using a particular emissivity profile.
The second graph is a log version of the first graph and a simple line fit tells us that Flux ≅ 4.85 x 10-7 T3.55.

Which used this emissivity profile:

It isn’t possible using anything remotely like an atmospheric emissivity profile to get Flux ∝T.
(The proof of this is left as an exercise for the interested student).

You can relate flux to temperature – using Planck’s law and the calculated emissivity for the atmosphere.

If you manage to make any other connection then I recommend devoting a paper just to that.

“If your special form of the virial theorem has radiant flux = kinetic energy then:”

Hold on there SoD. My first reaction was ‘well what about E = m, c^2’. Then I thought of ‘The Principle of Equivalence’ and googled.

I now realise how I, as an engineer, rely heavily on ‘Equivalence’ and a pertinent version of ‘The Virial Theorem’!

I’m happy that the ‘health and safety commission’ (HSC) here in the UK only permit ‘good practice’ for practising engineers (and I’m not currently in practise) because without this input engineers would be faced with the ‘proof of provenance’ that current science demands. From your writings here, it seems that the ‘science’ isn’t decided. That worries me. While googling I turned this up:

My personal point is that I’m glad I’m not a scientist and just a, run of the mill, ‘universal millwright type’ (pardon the pun), engineer.

My point of reasoning for the link to that paper is to show that the Virial Theorem ‘alone’ isn’t enough for an accurate demonstration of phenomenological energy transition. All the supporting formulae need to be integrated into it as well (see “3 General Covariance and the Weight of Light”) to provide the required ‘degrees of freedom’ that a representative ‘model’ is expected to mimic. Thus, I think you’ll need Ferenc’s ‘notes’.

See the graph at cell AO29.
The top straight line is equation 7:
Su – OLR +Ed -Eu = OLR. (I replace Fo with OLR)

The best fit of the HARTCODE NOAA data has a lot of scatter, but the slope of 0.9696 is very close to 1.

There has been a lot of criticism of equation 7. It is an empirical relationship. The equation is not a true conservation of energy equation.

Equation 7 is not used in M2010. The flux density ratio equation, Equ 11 of M2010 is derived using only the Ed/Eu = 5/3 equation 10 and the Aa = Ed (approximately) equation 5. The graph of equation 10 shows that Ed/Eu is constant with changing Su, but the value is closer to 1.645 rather than 1.6667.

The 95% confidence interval of the slope of your equation 10 plot is rather large, 0.69-1.25. The square of the correlation coefficient R is 0.45. The residual plot is clear evidence of non-linearity. Thirteen out of the first 14 residuals are positive and so are 9 of the last 14. 27 of the middle 33 residuals are negative. That’s proof only that x and y are somewhat correlated. It is extremely weak evidence at best that the implied relationship is correct.

Considering that the range of OLR values is less than 2% of the median, it’s a rather large stretch to imply that this is a general relationship that can be applied over a larger range. In fact, it’s a large stretch to assume that actual OLR, as opposed to calculated OLR, varied over that large a range for the time period, given the known large uncertainty in the NCEP/NCAR reanalysis data.

If we go back to the incorrect equations of M2007, the relationship of equation 7 only holds for τ > 3. Ed/Eu is only constant if τ is constant. For a purely radiative gray atmosphere, Ed/Eu doesn’t reach a value of 5/3 until τ = 2.92.

– The paper by Toth, as well as the more general approach taken in the paper he references by Pacheco and Sañuco, show that a careful application of the Virial Theorem to the planetary-atmosphere situation results in:

(2/3) =
relating the kinetic energy to the potential energy in a homogeneous gravitational field.

(2/5) =
for a diatomic molecular gas, if you include rotational energy in the kinetic energy.

– The result attributed by Miskolczi to the Virial Theorem is:
2 =
This is inconsistent with the above results.

– In any case, the application Miskolczi makes of this equation is somewhat mysterious: relating the ratio of two energy fluxes to the ratio of the gravitational and kinetic energies, for no very clear reason.